Note that \(\begin{bmatrix}\mathbf {\Phi }&\mathbf {\Gamma }\end{bmatrix}\)  internally stabilizes \(\begin{bmatrix}-I_m&\mathbf {G}^\top \end{bmatrix}\hspace{-2.84526pt}\phantom{.}^\top \)  if and only if (see Lemma 5.3 of [1]} for the continuous-time positive feedback case) all the entries of the following TFM
\(\widetilde{\mathbb {H}}\overset{def}{=}\footnotesize \begin{bmatrix}\phantom{-}I_m\\-I_m\\\phantom{-}\mathbf {G}\end{bmatrix}\left(I_m+\begin{bmatrix}\mathbf {\Phi }&\mathbf {\Gamma }\end{bmatrix}\begin{bmatrix}-I_m\\\phantom{-}\mathbf {G}\end{bmatrix}\right)^{-1}\begin{bmatrix}I_m&\mathbf {\Phi }&\mathbf {\Gamma }\end{bmatrix}\normalsize \) 
